Nagelsmann, whose sacking in March 2023 led to Tuchel's arrival, was tipped to return to Bayern, but has now extended his contract with Germany until 2026
Italian Claudio Ranieri won the coach of the year award on Monday for leading outsiders Leicester City to the English Premier League title.
The 31-year-old Ronaldo was rewarded for a year in which he won the Champions League with his club Real Madrid and then Euro 2016 with Portugal.
Argentine Messi was second in the vote and Frenchman Antoine Griezmann third.
"I'm very happy and I can say that 2016 was the best year of my career," said Ronaldo as he was presented the award by FIFA president Gianni Infantino.